Average Error: 0.9 → 0.1
Time: 3.4s
Precision: 64
\[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
↓
\[\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\]
\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}
↓
\frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)
double f(double re, double im) {
        double r27637 = im;
        double r27638 = re;
        double r27639 = atan2(r27637, r27638);
        double r27640 = 10.0;
        double r27641 = log(r27640);
        double r27642 = r27639 / r27641;
        return r27642;
}
↓
double f(double re, double im) {
        double r27643 = 1.0;
        double r27644 = 10.0;
        double r27645 = log(r27644);
        double r27646 = sqrt(r27645);
        double r27647 = r27643 / r27646;
        double r27648 = im;
        double r27649 = re;
        double r27650 = atan2(r27648, r27649);
        double r27651 = sqrt(r27647);
        double r27652 = r27650 * r27651;
        double r27653 = sqrt(r27651);
        double r27654 = r27652 * r27653;
        double r27655 = r27654 * r27653;
        double r27656 = r27647 * r27655;
        return r27656;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.9

    \[\frac{\tan^{-1}_* \frac{im}{re}}{\log 10}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.9

    \[\leadsto \frac{\tan^{-1}_* \frac{im}{re}}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

  4. Applied *-un-lft-identity0.9

    \[\leadsto \frac{\color{blue}{1 \cdot \tan^{-1}_* \frac{im}{re}}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

  5. Applied times-frac0.8

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\tan^{-1}_* \frac{im}{re}}{\sqrt{\log 10}}}\]
  6. Using strategy rm

  7. Applied div-inv0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\tan^{-1}_* \frac{im}{re} \cdot \frac{1}{\sqrt{\log 10}}\right)}\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\tan^{-1}_* \frac{im}{re} \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\right)\]

  10. Applied associate-*r*0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right)}\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.8

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\color{blue}{\sqrt{\frac{1}{\sqrt{\log 10}}} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}}}\right)\]

  13. Applied sqrt-prod0.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \color{blue}{\left(\sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}} \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)}\right)\]

  14. Applied associate-*r*0.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)}\]

  15. Final simplification0.1

    \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \left(\left(\left(\tan^{-1}_* \frac{im}{re} \cdot \sqrt{\frac{1}{\sqrt{\log 10}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right) \cdot \sqrt{\sqrt{\frac{1}{\sqrt{\log 10}}}}\right)\]

Reproduce

herbie shell --seed 2019354 
(FPCore (re im)
  :name "math.log10 on complex, imaginary part"
  :precision binary64
  (/ (atan2 im re) (log 10)))